# modiﬁed stochastic gradient based parameter estimation algorithm for dual-rate sampled-data systems

In this paper, we propose a novel identiﬁcation algorithm for a class of dual-rate sampleddata systems whose input–output data are measured by two different sampling rates. A polynomial transformation technique is employed to derive a mathematical model for such dual-rate systems. The proposed modiﬁed stochastic gradient algorithm has faster convergence rate than stochastic gradient algorithms for parameter identiﬁcation using the dual-rate input–output data. Convergence properties of the algorithm are analyzed. Finally, illustrative and comparison examples are provided to verify the effectiveness and performance improvement of the proposed method.

Multirate systems, in which the input and output signals have different sampling rates, can ﬁnd many engineering applications, e.g., in digital signal processing [1–3], communications [4], sensor networks [5], process control and estimation [6], to name a few. The problem of parameter estimation is of paramount importance in the analysis and design for multirate systems. Therefore, many works have been devoted to the parameter estimation of multirate systems. To develop an appropriate model structure that is consistent with all different input/output sampling rates is the ﬁrst step of parameter estimation. Generally, there are two main methods to transform the multirate model: • Lifting technique [7–10]. The lifting technique or blocking technique is widely used in dealing with multirate systems, including irregularly sampled-data systems [11,12]. By using a lifting operator, the fast input and slow output data are collected and a state-space model can be established. • Polynomial transformation technique [13,14,16]. This method can derive a multirate model that directly utilize all available data: Both the fast input and slow output data. With these two model transformation techniques, existing methods for multirate system parameter estimation include subspace algorithms [15], stochastic gradient (SG) algorithms [16], and least squares (LS) methods [13]. In [15], Li et al. used the least squares algorithm to estimate the parameters of the lifting state space models, then derived a fast single-rate model. However, subspace algorithms are oﬄine procedure and they are sensitive to the order modeling errors. The consistency of the SG algorithm for dual-rate sampled data systems was analyzed by using the polynomial transformation method [16]. Further, an LS algorithm was applied to effectively identify systems with noises in [13]. But it is known that

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